Equations (1) and (4) together give the energy of a configuration of the plate which is determined by $\epsilon x(x,y),\u2009\epsilon y(x,y),\u2009\epsilon xy(x,y)$, and *w*(*x*, *y*) (subject to the compatibility constraint equation (3)). It is instructive to make some estimates about the effect of fluctuations before we proceed further. From Eq. (1), we see that $Es\u223cYA\epsilon 2/2$, where *ε* represents in-plane strains, and $Eb\u223cKbA\kappa 2/2$, where *κ* represents a curvature. Using the equipartition theorem of statistical mechanics, we can estimate the mean-square fluctuations in these quantities as $\u2329\epsilon 2\u232a=kBT/YA$ and $\u2329\kappa 2\u232a=kBT/KbA$. If *h* is the thickness of our plate, then the ratio of mean-square fluctuations in the bending strain to in-plane strain is $\u2329\epsilon 2\u232a/\u2329\kappa 2\u232ah2=Kb/Yh2$. For a graphene sheet, *Y* ≈ 1 TPa × 0.3 nm, *K*_{b} ≈ 10^{−19} N·m, and *h* ≈ 0.3 nm [28], so that $Kb/Yh2\u224810\u22123$, and thus, in-plane strain fluctuations can be neglected in comparison to out-of-plane bending fluctuations. Due to the very high in-plane stretching modulus, we can also neglect the term $F(\epsilon x+\epsilon y)$ in Eq. (4) above. In the end, we assume that the plate has constant in-plane strains (no fluctuation) due to hydrostatic tension *F*, which contributes a constant term *C* to the expression for energy below
Display Formula

(5)$E=Eb+Ef+C=\u222cdxdy[Kb2(w,xx+w,yy)2+KG(w,xxw,yy\u2212w,xy2)+F(w,x22+w,y22)]+C$